Entanglement of assistance and coherence of assistance
A state is called incoherent if the density matrix \(\rho \)  is diagonal in the fixed reference basis \(\lbrace |i\rangle \rbrace \) , \(\rho =\sum _i p_i |i\rangle \langle i|\)  with \(p_i\)  the probabilities. Otherwise the state is called coherent. Obviously, coherence is basis dependent. A completely positive trace preserving map \(\Lambda \)  acting as \(\Lambda (\rho )=\sum _l K_l\rho K_l^\dagger \)  is said to be an incoherent operation if all the Kraus operators \(K_l\)  map incoherent states to incoherent states [1]}. A coherence measure \(C(\rho )\)  should satisfy [1]}: (1) \(C(\rho )\ge 0\)  with \(C(\rho )=0\)  if and only if \(\rho \)  is incoherent. (2) \(C(\rho )\)  is nonincreasing under incoherent operations \(\Lambda \) , \(C(\rho )\ge C(\Lambda (\rho ))\) . (3) \(C(\rho )\)  is nonincreasing on average under selective incoherent operations, \(C(\rho )\ge \sum _l q_l C(\rho _l)\) , with \(q_l=tr(K_l \rho K_l^\dagger )\)  and \(\rho _l=K_l \rho K_l^\dagger /q_l\) . (4) \(C(\rho )\)  is a convex function on the density matrices, \(C(\sum _j p_j \rho _j)\le \sum _j p_j C( \rho _j)\) .
